Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes

Abstract

The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo's navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The wind Finslerian structure is described in terms of two (conic, pseudo) Finsler metrics, one with a convex indicatrix and the other with a concave one. However, the spacetime viewpoint for the wind Riemannian case gives a useful unified viewpoint. A thorough study of the causal properties of such a spacetime is carried out in Finslerian terms. Randers-Kropina metrics appear as the Finslerian counterpart to the case of an SSTK when the Killing vector field is either timelike or lightlike. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary stationary wind, metric-type properties (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat's principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK spacetimes and Zermelo's navigation, are also provided.

AMSLaTex, 107 pages, 14 figures, one appendix; v6: final version; it contains some minor modifications and a new last section about related recent works. To appear in Memoirs AMS

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Cited by in corpus (19)

• On the definition and examples of cones and Finsler spacetimes
• Extended Hamiltonian Formalism and Lorentz-Violating Lagrangians
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Chains in CR geometry as geodesics of a Kropina metric
• Application of planar Randers geodesics with river-type perturbation in search models
• Geodesics on strong Kropina manifolds
• Zermelo navigation in pseudo-Finsler metrics
• A note on time-optimal paths on perturbed spheroid
• Anisotropic connections and parallel transport in Finsler spacetimes
• A Spacetime Geometry picture of Forest Fire Spreading and of Quantum Navigation
• A note on the Sagnac effect in general relativity as a Finslerian effect
• On Global Hyperbolicity of Spacetimes: Topology Meets Functional Analysis
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• A general model for time-minimizing navigation on a mountain slope under gravity
• Causal ladder of Finsler spacetimes with a cone Killing vector field
• On Generalized Matsumoto Metrics with a Special $\pi$-form